Integrated microwave-to-optical single-photon transducer with strain-induced electro-optic material

ABSTRACT

Transducers and methods of making the same include a substrate having a cavity with a diameter that supports whispering gallery modes at a frequency of an input signal. A focusing structure in the cavity focuses the electric field of the input signal. A resonator directly under the focusing structure has a crystalline structure that generates an electro-optic effect when exposed to electrical fields. An electric field of the input signal modulates an output signal in the resonator via the electro-optic effect.

BACKGROUND

Technical Field

The present invention relates to conversion between a microwave and anoptical domain and, more particularly, to a transducer to convertsingle-photon microwave signals to optical signals.

Description of the Related Art

Various communication protocols rely on optical fibers because of theirlow loss, high bandwidth, low background noise, and the ease of routing.Optical fibers can also be used for sending quantum information in theform of single photons or coherent states. On the other hand, manyviable quantum processing architectures operate at microwavefrequencies. The high amplitude stability of microwave structures allowsprecise controls on quantum bits (qubits) that enable high-fidelity gateoperations. However, microwave photons are more difficult to use forlong-range communication purposes, due to high thermal background noiseand high loss when such signals propagate in waveguides.

Existing approaches to converting between microwave signals and opticalsignals are complicated, difficult to implement in solid state systems,or difficult to optimize. Some existing transducers use electro-opticcrystalline optical resonators to perform microwave-to-opticalconversion. One of the problems of using such resonators is that othercoexisting non-linear properties, such as pyro-electricity andpiezo-electricity, impede the microfabrication processes of themicrowave resonator. Another problem is that microfabrication canlikewise contaminate the crystalline electro-optic optical resonatorsand reduce the quality factor. These resonators also have a highmicrowave loss and are difficult to align at cryogenic temperatures.

SUMMARY

A transducer includes a substrate having a cavity with a diameter thatsupports whispering gallery modes at a frequency of an input signal. Afocusing structure in the cavity focuses the electric field of the inputsignal. A resonator directly under the focusing structure has acrystalline structure that generates an electro-optic effect whenexposed to electrical fields. An electric field of the input signalmodulates an output signal in the resonator via the electro-opticeffect.

A quantum computing device includes a qubit configured to provide afirst signal at a first frequency. A transducer coupled to the qubit andincludes a substrate having a cylindrical cavity with a diameter thatsupports whispering gallery modes at the first frequency. There is acentral pin in the cavity. A resonator is positioned directly under thecentral pin. The resonator has a crystalline structure that generates anelectro-optic effect when exposed to electrical fields. An electricfield of the input signal modulates a second signal at a secondfrequency in the resonator via the electro-optic effect. A waveguide isoptically coupled to the resonator and is configured to convey themodulated second signal away from the resonator.

A method for forming a transducer includes fabricating a resonator on afirst substrate, resonant at a first frequency, by depositing astraining material on a resonator material to strain a crystallinestructure of the resonator material to generate an electro-optic effectwhen exposed to electrical fields. A second substrate is fabricated witha cavity. The cavity has a diameter that supports whispering gallerymodes at a second frequency. The second substrate is aligned over thefirst substrate such that a focusing structure in the microwave cavityaligns with the optical resonator.

These and other features and advantages will become apparent from thefollowing detailed description of illustrative embodiments thereof,which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The disclosure will provide details in the following description ofpreferred embodiments with reference to the following figures wherein:

FIG. 1 is a cross-sectional diagram of a microwave-to-optical transducerin accordance with the present principles;

FIG. 2 is a top-down diagram of a portion of a microwave-to-opticaltransducer in accordance with the present principles;

FIG. 3 is a bottom-up diagram of a portion of a microwave-to-opticaltransducer in accordance with the present principles;

FIG. 4 is a detailed cross-sectional diagram of a portion of amicrowave-to-optical transducer in accordance with the presentprinciples;

FIG. 5 is a schematic diagram of a microwave-to-optical transducer inaccordance with the present principles;

FIG. 6 is a detailed cross-sectional diagram of a strain-inducedelectro-optic optical resonator in accordance with the presentprinciples;

FIG. 7 is a block/flow diagram of a method of fabricating amicrowave-to-optical transducer in accordance with the presentprinciples;

FIG. 8 is a top-down diagram of a portion of an alternativemicrowave-to-optical transducer in accordance with the presentprinciples; and

FIG. 9 is a detailed cross-sectional diagram of a portion of analternative microwave-to-optical transducer in accordance with thepresent principles.

DETAILED DESCRIPTION

Embodiments of the present invention provide coupling betweensingle-photon microwave signals and single-photon infrared/opticalsignals via the electro-optic effect using superconducting microwave andoptical cavities. Each cavity incorporates an electro-optic material,with the electro-optic effect being induced by a straining material.Coupling takes place at the quantum level, with signal levels beingabout a single photon. The present embodiments may be implemented on onechip that may be fabricated using standard semiconductor fabricationprocesses.

Referring now to the drawings in which like numerals represent the sameor similar elements and initially to FIG. 1, a cross-sectional view of amicrowave-to-optical transducer 100 is shown. A bottom substrate 102 isshown as having, e.g., a quantum computing device 104 (a “qubit”) thatprovides, e.g., single-photon level microwave signals along asuperconducting channel 106 to a transducing cavity 130. It isspecifically contemplated that the bottom substrate 102 may be formedfrom silicon, but any other appropriate substrate material may be usedin its place. After converting the microwave signal to an optical signalin the transducing cavity 130, the optical signal couples with awaveguide 108 and is transmitted to its destination. The cavity 130 iscapacitively coupled to the superconducting channel 106, which may beeither a microwave transmission line, or to other resonating structures(e.g., the qubit 104 itself).

A top substrate 120 includes a cylindrical cavity 112 and a central pin114. In one embodiment, the cavity may have a radius of about 2.5 mm andthe central pin 114 may have a radius of about 2 mm and a height ofabout 2 mm. It is specifically contemplated that the top substrate 120may be formed from silicon, but any other appropriate substrate materialmay be used in its place. The sidewalls of the cavity 112 and thecentral pin 114 are coated with a superconducting film. The cavity 112joins with a similar cavity 111 on the bottom substrate 102 to form amicrowave resonator 122, which is connected to ground. The bottom cavity111 has an exemplary depth of 0.67 mm and an exemplary radius of 1.98nm. It should be noted that the top substrate 120 should not come intocontact with the bottom substrate, at least in regions withsuperconducting films or the channel 106, to prevent damage to thosestructures. The central pin 114 approaches, without touching, an opticalresonator 110 on a pedestal 115 in the bottom substrate 102. It isspecifically contemplated that the optical resonator 110 is formed fromsilicon and silicon-germanium, with the silicon-germanium providing astrain on the silicon material. In one embodiment, the optical resonator110 may have a radius of about 2 mm and a thickness of about 0.1 mm.This strain creates the electro-optic effect in the silicon as itdeforms the crystalline structure of the silicon.

The integrated design of the present embodiments minimizes alignmenterrors between the optical resonator 110 and the waveguide 108 as thecoupling between the optical resonator 110 and the waveguide 108 isdefined by microfabrication. Such alignment errors would otherwise occurif the optical couplers were not integrated into the device, for examplein systems that use prisms for coupling. In particular, in a cryogenicenvironment at millikelvin temperatures, misalignment errors due todifferent thermal expansion coefficients of the different materials canbe reduced or avoided entirely.

During operation, microwave signals from the qubit 104 couple to themicrowave resonator 122 , where a standing wave forms on the outer andinner circumferences of the cavity, with strong fields at the boundariesand negligible field strength in the middle of the cavity. Thesuperconducting film creates a low-loss resonator with a very high Q. Atthe junction of the central pin 114 with the optical resonator 110, thefields of the microwave modes modulate an optical signal in the opticalresonator 110. With the aid of optical pump signals applied to theoptical resonator, a microwave signal can be converted into an opticalsignal at a single photon level.

In one embodiment, the microwave resonator can be formed in an on-chip,transmission-line cavity or a coplanar waveguide cavity. A center pin ora high-voltage electrode of a transmission-line cavity or a coplanarwaveguide cavity has a circular shape that can deliver a microwavesignal to the optical resonator.

The optical resonator 110 may be formed in the shape of a disc, asshown, or as a ring, in both embodiments supporting whispering gallerymodes at multiple frequencies. The diameter of the optical resonator 110is selected to provide three modes at frequencies ω_(op)-ω_(q) for ared-sideband, ω_(op) for a carrier, and ω_(op)+ω_(q) for a blue-sideband, with ω_(q) being the microwave frequency of the microwaveresonator 122. In one embodiment, ω_(op)/2π may be about 193 THz (1550nm wavelength) and ω_(q) may be about 10 GHz. This embodiment may beachieved by choosing the free spectral range to be ω_(q), which isdetermined by the refractive index and diameter of the optical resonator110. Using the sideband modes, a three-wave mixer is realized thatcouples microwave photons and optical photons.

Referring now to FIG. 2, a top-down view of the bottom substrate 102 isshown. The superconducting qubit 104 and the superconducting channel 106are formed in or on the substrate 102. The lower cavity 111 is formed inthe bottom substrate 102 by any appropriate process, including, e.g.,micromachining or etching. A superconducting film is deposited over theinternal surfaces of the lower cavity 111. The superconducting film mayinclude, for example, aluminum, niobium, titanium, indium, or any othermaterial or alloy that demonstrates superconducting properties in adesired temperature range. The superconducting film may be deposited by,e.g., sputtering or by thermal evaporation in a vacuum chamber. Theoptical resonator 110 is formed from, e.g., a saw-tooth silicon disc orring with a layer of silicon-germanium to provide strain to thecrystalline structure of the silicon, as described in further detailbelow. The optical waveguide 108 couples with the optical resonator 110to transmit the optical signal off-chip and, in one embodiment, theoptical waveguide 108 is positioned less than one micron away from theoptical resonator 110 to promote coupling.

Referring now to FIG. 3, a bottom-up view of the top substrate 120 isshown. The upper cavity 112 is formed in the top substrate 120 by anyappropriate process, including, e.g., micromachining or etching. Thecentral pin 114 is formed by the micromachining process as well, and asuperconducting film is deposited over the surfaces of the upper cavity112 and the central pin 114. A ridge 302 is formed along the outer edgeof the facing circle of the central pin 114. The ridge 302 concentratesthe fields of the whispering gallery modes along this edge for couplingwith the optical resonator 110. The ridge 302 may be formed by anyappropriate process, including, micromachining or etching. When the topsubstrate 120 is placed above the bottom substrate 102, the ridge 302aligns with the outer edge of the optical resonator 110.

Referring now to FIG. 4, a more detailed cross-sectional view of theconnection between the pin 114 and the optical resonator 110 is shown.The ridge 302 is positioned slightly above the optical resonator, withsmall gap between the two structures to prevent plasmonic loss of theoptical signal through the superconducting film. A small portioninternal portion of the face of the central pin 114 is cut away, with acutaway depth of about 0.5 mm and a cutaway radius of about 1.9 mm. Inaddition, superconducting surfaces 402 are shown with a heavier lineweight, having had a superconducting film deposited on them.

It should be noted that the central pin 114 is also recessed withrespect to the sidewalls of the cavity 112. The depth of the recessprovides room for the optical resonator 110 as well as a smalladditional gap to prevent plasmonic modes between the pin 114 and theresonator 110. The cavity 112 and gap may be filled with air or may bein a vacuum or an appropriate inert gas.

It is to be understood that the present invention will be described interms of a given illustrative architecture having a wafer; however,other architectures, structures, substrate materials and processfeatures and steps may be varied within the scope of the presentinvention.

It will also be understood that when an element such as a layer, regionor substrate is referred to as being “on” or “over” another element, itcan be directly on the other element or intervening elements may also bepresent. In contrast, when an element is referred to as being “directlyon” or “directly over” another element, there are no interveningelements present. It will also be understood that when an element isreferred to as being “connected” or “coupled” to another element, it canbe directly connected or coupled to the other element or interveningelements may be present. In contrast, when an element is referred to asbeing “directly connected” or “directly coupled” to another element,there are no intervening elements present.

A design for an integrated circuit chip may be created in a graphicalcomputer programming language, and stored in a computer storage medium(such as a disk, tape, physical hard drive, or virtual hard drive suchas in a storage access network). If the designer does not fabricatechips or the photolithographic masks used to fabricate chips, thedesigner may transmit the resulting design by physical means (e.g., byproviding a copy of the storage medium storing the design) orelectronically (e.g., through the Internet) to such entities, directlyor indirectly. The stored design is then converted into the appropriateformat (e.g., GDSII) for the fabrication of photolithographic masks,which typically include multiple copies of the chip design in questionthat are to be formed on a wafer. The photolithographic masks areutilized to define areas of the wafer (and/or the layers thereon) to beetched or otherwise processed.

Methods as described herein may be used in the fabrication of integratedcircuit chips. The resulting integrated circuit chips can be distributedby the fabricator in raw wafer form (that is, as a single wafer that hasmultiple unpackaged chips), as a bare die, or in a packaged form. In thelatter case the chip is mounted in a single chip package (such as aplastic carrier, with leads that are affixed to a motherboard or otherhigher level carrier) or in a multichip package (such as a ceramiccarrier that has either or both surface interconnections or buriedinterconnections). In any case the chip is then integrated with otherchips, discrete circuit elements, and/or other signal processing devicesas part of either (a) an intermediate product, such as a motherboard, or(b) an end product. The end product can be any product that includesintegrated circuit chips, ranging from toys and other low-endapplications to advanced computer products having a display, a keyboardor other input device, and a central processor.

Reference in the specification to “one embodiment” or “an embodiment” ofthe present principles, as well as other variations thereof, means thata particular feature, structure, characteristic, and so forth describedin connection with the embodiment is included in at least one embodimentof the present principles. Thus, the appearances of the phrase “in oneembodiment” or “in an embodiment”, as well any other variations,appearing in various places throughout the specification are notnecessarily all referring to the same embodiment.

It is to be appreciated that the use of any of the following “/”,“and/or”, and “at least one of”, for example, in the cases of “A/B”, “Aand/or B” and “at least one of A and B”, is intended to encompass theselection of the first listed option (A) only, or the selection of thesecond listed option (B) only, or the selection of both options (A andB). As a further example, in the cases of “A, B, and/or C” and “at leastone of A, B, and C”, such phrasing is intended to encompass theselection of the first listed option (A) only, or the selection of thesecond listed option (B) only, or the selection of the third listedoption (C) only, or the selection of the first and the second listedoptions (A and B) only, or the selection of the first and third listedoptions (A and C) only, or the selection of the second and third listedoptions (B and C) only, or the selection of all three options (A and Band C). This may be extended, as readily apparent by one of ordinaryskill in this and related arts, for as many items listed.

Referring now to FIG. 5, an abstract diagram of electro-optic modulationof an optical carrier is shown. An optical beam 508 is shown passingthrough semi-transparent, partially mirrored plate 502, through anelectro-optic modulator (EOM) region 506, and reflecting from a mirror504. The mirrors 502 and 504 form a Fabry-Perot cavity. In the presentembodiments, the EOM region 506 is a resonator at the opticalwavelengths. An inductor 514 and the plates of capacitor 512 form aresonator at a microwave frequency, with the output of qubit 510 beinginjected into the resonator through capacitor 516. As the microwavesignal oscillates in the resonator, charges build and switch oncapacitor plates 512 around the EOM region 506. These charges create anoscillating electric field that causes a phase shift in the opticalsignal.

The phase shift is caused by a change in the refractive index in the EOM506 caused by the external electric field, E_(j). This change ischaracterized as:

${\Delta\; n} = {{- \frac{1}{2}}n^{3}r_{ij}E_{j}}$where n is the refractive index of the medium of the EOM 506 and r_(ij)is the electro-optic coefficient. The phase shift is characterized as:

${\Delta\;\phi} = {{\Delta\;{kL}} = {\frac{\omega_{a}}{{c/\Delta}\; n}L}}$where L is the inductance of inductor 514 and ω_(a) is the frequency ofthe optical signal. A change in the frequency is characterized by:

${\Delta\;\omega_{a}} = {\frac{\Delta\;\phi}{\tau} = {{\frac{\omega_{a}}{{c/\Delta}\; n}\frac{L}{\tau}} = {{\omega_{a}\frac{\Delta\; n}{n}} = {{- \frac{1}{2}}\omega_{a}n^{2}r_{ij}E_{j}}}}}$where τ is the optical round-trip time and c is the speed of light. Theindices i and j are the indices of the electro-optic material crystalaxis.

This embodiment specifically makes use of the Pockels effect, where theresonant frequency of the optical resonator 110 is modulated using themicrowave field from the microwave cavity 112. The electro-optic effectis caused in the optical resonator 110 by depositing a strainingmaterial that breaks the crystal symmetry of a substrate. The couplingbetween the microwave signal and the optical signal is described by acoupling Hamilton:Ĥ′=hg({circumflex over (b)} ⁺ +{circumflex over (b)})(â ⁻ ⁺ +â ⁺ +â ₊⁺)(â ⁻ +â+â ⁺)where â⁻(â⁻ ⁺), â(â⁺), and â₊(â₊ ⁺) are the annihilation (creation)operators for red-sideband, carrier, and blue-sideband modes in theoptical cavity 110 respectively, {circumflex over (b)}({circumflex over(b)}⁺) is the annihilation (creation) operator for the microwave photonsof the qubit 104, and g is the coupling strength between the optical andmicrowave photons. After applying the rotating-wave approximation, theHamiltonian Ĥ=Ĥ₀+Ĥ_(C) of the electro-optic device 100, including thered- and blue-sideband modes, becomes:Ĥ ₀ hω ⁻ â ⁻ ⁺ â ⁻ +hω _(op) â ⁺ â+hω ₊ â ₊ ⁺ â ₊ +hω _(q) {circumflexover (b)} ⁺ {circumflex over (b)}Ĥ _(C) =hg({circumflex over (a)}±{circumflex over (a)}{circumflex over(b)}⁺+{circumflex over (a)}_{circumflex over (a)}⁺{circumflex over(b)}+{circumflex over (a)}₊ ⁺{circumflex over (a)}{circumflex over(b)}+{circumflex over (a)}₊{circumflex over (a)}⁺{circumflex over (b)}⁺)where ω_(op) is the optical carrier frequency, ω⁻ and ω₊ are the red andblue sideband frequencies, and ω_(q) is the microwave frequency of thequbit 104.

The coupling Hamiltonian, Ĥ_(C), shows the three-wave mixing among theoptical photons at the carrier and the sidebands and the microwavephotons of the superconducting qubit 104. By applying a strong pump toneat ω₊=ω_(op)+ω_(q), the operators â⁺ can be replaced by the classicaldrive a⁺ (c-number), where |a⁺|² represents the average number of pumpphotons at ω⁺, which provides an effective coupling rate between thequantum microwave node {circumflex over (b)} and the fundamental opticalmode â at a rate Ω_(R)=g|a₊|. In one embodiment, with realisticparameters, this rate may be about 10 MHz, with a coupling strength ofg˜10 kHz and a=1000 corresponding to 10⁶ photons in the resonators. Thissets an upper bound on the speed of the communication channel. Ingeneral the coupling strength can be estimated as:

$\frac{g}{2\;\pi} \equiv {{{- \frac{1}{2}}f_{a}n^{2}r\frac{V_{ZPV}}{d}\text{∼}0.1} - {100\mspace{14mu}{kHz}}}$where V_(ZPV) is the superconducting cavity zero-point voltage (having arange of about 0.1 μV to about 1 μV), d is the thickness of the opticalresonator 110 (having a range of about 1 μm to about 100 μm), f_(a) isan optical communications frequency (e.g., about 193 THz), n is thenumber of pump photons, and r is the electro-optic coefficient of theelectro-optic material. It should be noted that the pump signal can beprovided through the optical waveguide 108 described above.

Referring now to FIG. 6, a cross-sectional view of detail on thestructure of an optical resonator 110 is shown. The substrate 102 has aring 602 of additional material, e.g., silicon, formed on it. Thematerial of the ring 602 is patterned on a top surface, with anothermaterial 604, e.g., silicon germanium, being deposited in the gaps. Theadditional material 604 is selected to have a different latticestructure than that of the ring 602, which causes a strain in thelattice structure of the ring 602. It is this strain that makes the ring602 susceptible to the electro-optic effect. The straining material 604may deposited by any appropriate deposition process including, e.g.,chemical vapor deposition, physical vapor deposition, and atomic layerdeposition.

It is contemplated that other embodiments of an optical resonator 110may be employed. As noted above, the ring 602 is only one structure, anda disc embodiment may be used instead as long as it supports whisperinggallery modes at the optical frequencies in question. In addition,different materials may be used. The above-described embodiment usesstrain in the crystalline lattice structure of the resonator 110 tocreate the Pockels electro-optic effect, but it should be noted thatsome materials that naturally lack crystalline inversion symmetry alsoexhibit this effect and may be used instead. The optical resonator 110may be formed by any appropriate fabrication technique, includingmachining, micro-fabrication, etching, etc.

Referring now to FIG. 7, a method of forming a microwave-to-opticaltransducer is shown. Block 702 fabricates the optical resonator 110. Inparticular, block 702 forms the optical resonator in or on the bottomsubstrate 102 by, e.g., depositing resonator material in block 704 (or,alternatively, etching the resonator material from the bulk substrate102 ), patterning the resonator surface in block 706 as described aboveto form ridges, and depositing a straining material 604 in block 708 tocreate a strain in the crystalline structure of the optical resonator110. Alternatively, block 702 can fabricate the optical resonator 110from a material that naturally exhibits the electro-optic (Pockels)effect.

Block 710 constructs the microwave cavity in the top substrate 120.Block 712 machines the microwave cavity 112 in the top substrate 120 byany appropriate micro-fabrication technique, including micro-machiningor etching. The microwave cavity 112 may be, for example, a microwavecoaxial cavity (as shown above), a microwave coplanar waveguide, amicrowave microstrip cavity, etc., and is formed with a smooth internalsurface at a diameter that supports whispering gallery modes at themicrowave frequency of the qubit 104. Block 714 forms the ridge 302 onthe face of the central pin 114 by, e.g., machining the surface of thepin 114 or by an etching process to concentrate the electric fields ofthe microwave signal onto the optical resonator 110. Block 716 forms asuperconducting film over the internal surface of the microwave cavity112 and the outer surface of the central pin 114.

Block 716 fabricates the qubit(s) 104 on the lower substrate 102. Itshould be noted that the qubit(s) 104 may be made with superconductingmaterial and may be integrated with the same substrate 102 as describedabove, or may be formed in a separate package and subsequently connectedor attached to the device.

Block 718 forms a waveguide in, e.g., the bottom substrate 102, thatcouples to the optical resonator 110 and provides communication ofmodulated signals from the optical resonator 110 to other devices on- oroff-chip. Block 720 forms a superconducting coupling path 106 thatcouples the qubit(s) 104 to the microwave electric fields in themicrowave cavity 112. The coupling path may include, e.g., a microwaveantenna or superconducting channel. Block 722 assembles the transducer,placing the top substrate 120 over the bottom substrate 102 and aligningthe central pin 114 of the microwave cavity 112 above the opticalresonator 110, such that electric fields from the whispering gallerymodes on the central pin 114 are applied to the optical resonator 110.

Referring now to FIG. 8, a top-down view of an alternative bottomsubstrate 802 is shown. The qubit 104, superconducting channel 106, andwaveguide 108 are shown as being placed similarly to the embodiment ofFIG. 2. However, instead of having a lower cavity 111 with the opticalresonator 110 being placed over top, this embodiment has a ring opticalresonator 804 placed directly on the bottom substrate 802 with no lowercavity at all. This embodiment may alternatively have a disc oscillatoras the optical resonator 804.

Referring now to FIG. 9, a more detailed cross-sectional view of theconnection between the pin 114 and the optical resonator 804 is shown.As can be seen, the resonator 804 rests directly on the bottom substrate102 and is shown in cross section directly under the ridge 302.

Having described preferred embodiments of an integratedmicrowave-to-optical single-photon transducer (which are intended to beillustrative and not limiting), it is noted that modifications andvariations can be made by persons skilled in the art in light of theabove teachings. It is therefore to be understood that changes may bemade in the particular embodiments disclosed which are within the scopeof the invention as outlined by the appended claims. Having thusdescribed aspects of the invention, with the details and particularityrequired by the patent laws, what is claimed and desired protected byLetters Patent is set forth in the appended claims.

The invention claimed is:
 1. A method for forming a transducer,comprising: fabricating an optical resonator on a first substrate,resonant at an optical frequency, by depositing a straining material onan optical resonator material to strain a crystalline structure of theoptical resonator material to generate an electro-optic effect whenexposed to electrical fields; fabricating a second substrate having amicrowave cavity with a diameter that supports whispering gallery modesat a microwave frequency; and aligning the second substrate over thefirst substrate such that a focusing structure in the microwave cavityaligns with the optical resonator.
 2. The method of claim 1, whereinfabricating the optical resonator comprises: patterning the opticalresonator to form grooves; and depositing the straining material in thegrooves.
 3. The method of claim 1, wherein the depositing the strainingmaterial includes depositing silicon germanium in the grooves.
 4. Themethod of claim 1, wherein fabricating the second substrate having themicrowave cavity includes forming a cylindrical cavity such that thefocusing structure is a center pin that is coaxial with the cylindricalcavity.
 5. The method of claim 1, further comprising depositing asuperconducting film in the microwave cavity.
 6. The method of claim 5,wherein depositing the superconducting film in the microwave cavityincludes depositing the superconducting film directly on an innersurface of the microwave cavity and on an outer surface of the focusingstructure.
 7. The method of claim 1, further comprising forming a ridgeon an outer edge of a surface of the focusing structure facing theoptical resonator.
 8. The method of claim 1, further comprising forminga second cavity underneath the optical resonator having a same diameteras the microwave cavity in the first substrate.
 9. The method of claim1, further comprising forming a microwave coupler in the first substrateto couple a qubit to microwave electric fields in the microwave cavity.10. The method of claim 1, further comprising forming a waveguide,optically coupled to the optical resonator, that is configured to conveya modulated optical signal away from the optical resonator.
 11. Themethod of claim 1, wherein fabricating the optical resonator includesfoaming an optical disc structure.
 12. The method of claim 1, whereinfabricating the optical resonator includes forming an optical ringstructure.
 13. The method of claim 1, wherein aligning the secondsubstrate over the first substrate includes positioning the focusingstructure directly on the optical resonator.
 14. The method of claim 1,wherein aligning the second substrate over the first substrate includespositioning the focusing structure above the optical resonator toprovide a gap between the focusing structure and the optical resonator.15. A method for forming a transducer, comprising: fabricating anoptical resonator on a first substrate, resonant at an opticalfrequency, by depositing a straining material on an optical resonatormaterial to strain a crystalline structure of the optical resonatormaterial to generate an electro-optic effect when exposed to electricalfields; fabricating a second substrate having a microwave cavity with adiameter that supports whispering gallery modes at a microwavefrequency; depositing a superconducting film directly on an innersurface of the microwave cavity and on an outer surface of a focusingstructure disposed in the microwave cavity; and aligning the secondsubstrate over the first substrate such that the focusing structure inthe microwave cavity aligns with the optical resonator.
 16. The methodof claim 15, wherein fabricating the optical resonator comprises:patterning the optical resonator to form grooves; and depositing thestraining material in the grooves.
 17. The method of claim 15, furthercomprising forming a ridge on an outer edge of a surface of the focusingstructure facing the optical resonator.
 18. The method of claim 15,further comprising forming a waveguide, optically coupled to the opticalresonator, that is configured to convey a modulated optical signal awayfrom the optical resonator.
 19. The method of claim 15, wherein aligningthe second substrate over the first substrate includes positioning thefocusing structure directly on the optical resonator.
 20. The method ofclaim 15, wherein aligning the second substrate over the first substrateincludes positioning the focusing structure above the optical resonatorto provide a gap between the focusing structure and the opticalresonator.